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\title{Notes on HyperNova}
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\author{arnaucube}
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\date{May 2023}
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\begin{document}
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\maketitle
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\begin{abstract}
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Notes taken while reading about Spartan \cite{cryptoeprint:2023/573}, \cite{cryptoeprint:2023/552}.
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Usually while reading papers I take handwritten notes, this document contains some of them re-written to $LaTeX$.
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The notes are not complete, don't include all the steps neither all the proofs.
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\end{abstract}
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\tableofcontents
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\section{CCS}
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\subsection{R1CS to CCS overview}
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\begin{itemize}
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\item[] R1CS instance: $S_{R1CS} = (m, n, N, l, A, B, C)$
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\item[] CCS instance: $S_{CCS} = (m, n, N, l, t, q, d, M, S, c)$
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\item[] R1CS-to-CCS parameters:\\
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$n=n,~ m=m,~ N=N,~ l=l,~ t=3,~ q=2,~ d=2$\\
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$M=\{A,B,C\}$, $S=\{\{0,~1\},~ \{2\}\}$, $c=\{1,-1\}$
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\end{itemize}
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Then, we can see that the CCS relation:
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$$\sum_{i=0}^{q-1} c_i \cdot \bigcirc_{j \in S_i} M_j \cdot z ==0$$
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where $z=(w, 1, x) \in \mathbb{F}^n$.
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In our R1CS-to-CCS parameters is equivalent to
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\begin{align*}
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&c_0 \cdot ( (M_0 z) \circ (M_1 z) ) + c_1 \cdot (M_2 z) ==0\\
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\Longrightarrow &1 \cdot ( (A z) \circ (B z) ) + (-1) \cdot (C z) ==0\\
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\Longrightarrow &( (A z) \circ (B z) ) - (C z) ==0
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\end{align*}
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which is equivalent to the R1CS relation: $Az \circ Bz == Cz$
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An example of the conversion from R1CS to CCS implemented in SageMath can be found at\\
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\href{https://github.com/arnaucube/math/blob/master/r1cs-ccs.sage}{https://github.com/arnaucube/math/blob/master/r1cs-ccs.sage}.
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\subsection{Committed CCS}
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$R_{CCCS}$ instance: $(C, \mathsf{x})$, where $C$ is a commitment to a multilinear polynomial in $s'-1$ variables.
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Sat if:
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\begin{enumerate}[i.]
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\item $\text{Commit}(pp, \widetilde{w}) = C$
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\item $\sum_{i=1}^q c_i \cdot \left( \prod_{j \in S_i} \left( \sum_{y \in \{0,1\}^{\log m}} \widetilde{M}_j(x, y) \cdot \widetilde{z}(y) \right) \right)$\\
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where $\widetilde{z}(y) = \widetilde{(w, 1, \mathsf{x})}(x) ~\forall x \in \{0, 1\}^{s'}$
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\end{enumerate}
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\subsection{Linearized Committed CCS}
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$R_{LCCCS}$ instance: $(C, u, \mathsf{x}, r, v_1, \ldots, v_t)$, where $C$ is a commitment to a multilinear polynomial in $s'-1$ variables, and $u \in \mathbb{F},~ \mathsf{x} \in \mathbb{F}^l,~ r \in \mathbb{F}^s,~ v_i \in \mathbb{F} ~\forall i \in [t]$.
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Sat if:
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\begin{enumerate}[i.]
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\item $\text{Commit}(pp, \widetilde{w}) = C$
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\item $\forall i \in [t],~ v_i = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_i(r, y) \cdot \widetilde{z}(y)$\\
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where $\widetilde{z}(y) = \widetilde{(w, u, \mathsf{x})}(x) ~\forall x \in \{0, 1\}^{s'}$
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\end{enumerate}
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\section{Multifolding Scheme for CCS}
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Recall sum-check protocol:\\
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\underline{$C \leftarrow <P, V(r)>(g, l, d, T)$}:\\ % TODO use proper <, >
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$T=\sum_{x_1 \in \{0,1\}} \sum_{x_2 \in \{0,1\}} \cdots \sum_{x_l \in \{0,1\}} g(x_1, x_2, \ldots, x_l)$
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$l$-variate polynomial g, degree $\leq d$ in each variable.
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let $s= \log m,~ s'= \log n$.
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\begin{enumerate}
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\item $V \rightarrow P: \gamma \in^R \mathbb{F},~ \beta \in^R \mathbb{F}^s$
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\item $V: r_x' \in^R \mathbb{F}^s$
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\item $V \leftrightarrow P$: sum-check protocol:\\
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$$c \leftarrow <P, V(r_x')>(g, s, d+1, \sum_{j \in [t]} \gamma^j \cdot v_j)$$
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where:\\
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\begin{align*}
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g(x) &:= \left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right) + \gamma^{t+1} \cdot Q(x)\\
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L_j(x) &:= \widetilde{eq}(r_x, x) \cdot \left( \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y) \right)\\
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Q(x) &:= \widetilde{eq}(\beta, x) \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y) \right) \right)
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\end{align*}
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\item $P \rightarrow V$: $\left( (\sigma_1, \ldots, \sigma_t), (\theta_1, \ldots, \theta_t) \right)$
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where
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$$\sigma_j = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y)$$
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$$\theta_j = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_2(y)$$
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\item V: $e_1 \leftarrow \widetilde{eq}(r_x, r_x')$, $e_2 \leftarrow \widetilde{eq}(\beta, r_x')$\\
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check:
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$$c = \left( \sum_{j \in [t]} \gamma^j e_1 \sigma_j + \gamma^{t+1} e_2 \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \sigma \right) \right)$$
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\item $V \rightarrow P: \rho \in^R \mathbb{F}$
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\item $V, P$: output the folded LCCCS instance $(C', u', \mathsf{x}', r_x', v_1', \ldots, v_t')$, where $\forall i \in [t]$:
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\begin{align*}
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C' &\leftarrow C_1 + \rho \cdot C_2\\
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u' &\leftarrow u + \rho \cdot 1\\
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\mathsf{x}' &\leftarrow \mathsf{x}_1 + \rho \cdot \mathsf{x}_2\\
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v_i' &\leftarrow \sigma_i + \rho \cdot \theta_i
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\end{align*}
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\item $P$: output folded witness: $\widetilde{w}' \leftarrow \widetilde{w}_1 + \rho \cdot \widetilde{w}_2$.
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\end{enumerate}
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\bibliography{paper-notes.bib}
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\bibliographystyle{unsrt}
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\end{document}
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